Immediate Inference

Of the foregoing relationships the most important are those known by the special names of contrariety and contradiction. Two propositions are called contraries when both cannot be true at the same time, but both may be false. SaP and SeP are related in this way, as the above account shows. Again two propositions are called contradictories when both cannot be true at the same time, but also not false at the same time, in other words, when one of them must be true and the others false. From the above account it will be seen that SaP and SoP are related in this way, also SeP and SiP. Two propositions are also contrary when they affirm contrary predicates, like "white" and "black" of the same subject; and contradictory when they affirm contradictory predi cates, like "white" and "non-white," of the same subject. It should be noted that SiP and SoP, which are commonly called sub-contraries, are related in a manner which is the precise re verse of that between contraries, for both can be true, and one of them must be true.

Eductions.

Given a proposition having for its terms S-P it always implies some proposition having for its terms S-P, which is called its obverse. Thus SaP, in virtue of the law of contra diction, implies since S cannot be both P and P, the S's which are P are not T. Similarly, SiP implies SoP. Again, SeP implies SaP, in virtue of the law of extended middle; must be either P or P, the S's that are not P must be P. Similarly SoP implies SiP. It should be noted that in each case if the obverse is obverted the original proposition is obtained. Thus SeP im plies, in virtue of the law of excluded middle, SaP, and so with the rest. Again, given a proposition having for its terms S-P it sometimes (but not always) implies a proposition having for its terms P-S, which is called its converse. Thus SeP implies PeS, for it asserts the mutual exclusion of S and P, and it makes no essential difference whether, looking at the relation from the point of view of S, we say SeP or, looking at it from the point of view of P, we say PeS.

Similarly

SiP implies PiS, for it expresses the fact that some things are both S and P (SP), and it makes no essential differ ence whether these things are described as some S's that are P or as some P's that are S's. SaP likewise expresses the fact that some things are both S and P, but whereas it tells us that these include every S it does not say that they include every P. Hence

it implies PiS, not PaS. To make SaP imply PaS would violate the above-mentioned rule about the distribution of terms, for P is not distributed in the premise SaP, but is distributed in PaS. In other words, SaP may be true even when PaS is not true, that is, when PoS is true (e.g., "All Englishmen are British subjects" but not "All British subjects are Englishmen," for "Some British subjects are not Englishmen"). Lastly, SoP has no converse, for in PoS the term S would be distributed, and it is not dis tributed in SoP; in other words PaS may be true at the same time as SoP (e.g., "Some British subjects are not Englishmen," yet "All Englishmen are British subjects," so that it would not be true to say that "Some Englishmen are not British subjects"). In the case of the converse, as in the case of the obverse, we get no further forward if we take the converse of the converse. It only brings us back to the original proposition or to something less. Thus the converse of PeS is SeP, of PiS it is SiP, even where the original was SaP.

The obverse and the converse are the fundamental types of eduction, but by combining obversion and conversion (that is by applying them alternately, beginning with obversion and going on to conversion, and so on alternately, or beginning with con version and going on to obversion and so on alternately), certain derivation eductions are obtainable. The following table sets out all the eductions and shows how they are obtained. The arrow means "implies," "(o)" means "by obversion," "(c)" "by conversion." All the eductions have special names which may just be mentioned here. Assuming that the terms of the original proposition are S-P, the obverse has the term; S-P; the converse has P-S; the obverted converse has P-S; the converted obverse or contraposi tion has P-S; the obverted contraposition has P-S; the inverse has s-P; the obverted inverse — There are some other types of immediate inference of which only one need be mentioned here as it is fairly common. A proposi tion containing a relative term in the predicate implies a correl ative proposition. Thus "S is north of P" implies "P is south of S"; "S is the parent of P" implies "P is the child of S." Symbolically, S is RP implies P is fl S, when R and H represent correlative terms.